General Theory -- Two-Dimensional Coordinate Systems. Rectangular Coordinates -- Circular Coordinates -- Elliptic Coordinates -- Parabolic Coordinates -- Three-Dimensional Coordinate Systems. Rectangular Coordinates -- Circular Cylinder Coordinates -- Elliptic Cylinder Coordinates -- Parabolic Cylinder Coordinates -- Spherical Polar Coordinates -- Prolate Spheroidal Coordinates -- Oblate Spheroidal Coordinates -- Parabolic Rotational Coordinates -- Conical Coordinates -- Ellipsoidal Coordinates -- Paraboloidal Coordinates -- Advanced Formulations. Differential-Geometric Formulation -- Quantum-Mechanical Particle Confined to the Neighborhood of Curves -- Quantum-Mechanical Particle Confined to Surfaces of Revolution -- Boundary Perturbation Theory -- Appendix A: Hypergeometric Functions -- Appendix B: Baer Functions -- Appendix C: Bessel Functions -- Appendix D: Lam̌ Functions -- Appendix E: Legendre Functions -- Appendix F: Mathieu Functions -- Appendix G: Spheroidal Wave Functions -- Appendix H: Weber Functions -- Appendix I: Elliptic Integrals and Functions.

Includes bibliographical references and index.

Innovative developments in science and technology require a thorough knowledge of applied mathematics, particularly in the field of differential equations and special functions. These are relevant in modeling and computing applications of electromagnetic theory and quantum theory, e.g. in photonics and nanotechnology. The problem of solving partial differential equations remains an important topic that is taught at both the undergraduate and graduate level.
The proposed book has a very comprehensive coverage on partial differential equations in a variety of coordinate systems and geometry, and their solutions using the method of separation of variables. The treatment includes complete details on going from the basic theory (including separability conditions not presented in introductory texts) to full implementation for applications. A very good choice of examples is inspired by the authors? research on semiconductor nanostructures and metamaterials and include modern applications like quantum dots.
The fluency of the text and the high quality of graphics make the topic easy accessible. The organization of the content by coordinate systems rather than by equation types is unique and offers an easy access.
The authors consider recent research results which have led to a much increased pedagogical understanding of not just this topic but of many other related topics in mathematical physics, and which? like the explicit discussion on differential geometry shows - yet have not been treated in the older texts. To the benefit of the reader, a summary presents a convenient overview on all special functions covered. Homework problems are included as well as numerical algorithms for computing special functions. Thus this book can serve as a reference text for advanced undergraduate students, as a textbook for graduate level courses, and as a self-study book and reference manual for physicists, theoretically oriented engineers and traditional mathematicians.

MA4300, PH2300 suitable for graduate level course; could serve as one of two main texts of a partial differential equations course.

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